
Since \(({x}_t-{x}^\star , {\lambda }_t-{\lambda }^\star ) \rightarrow {0}\)  as \(t\rightarrow \infty \) , and by the fact that \(a_t\rightarrow a\)  implies \(\frac{1}{t}\sum _{i=0}^{t-1}a_i\rightarrow a\)  for any sequence \(a_t\)  (known as Stolz–Cesàro theorem), it suffices to show that \((\sum _{i=0}^{t-1} {\bar{H}}_i - \nabla ^2f_i)/t\)  converges.
In fact, noting from Assumption REF () that \(\mathbb {E}[{\bar{H}}_i \mid \mathcal {F}_{i-1}] = \nabla ^2 f_i\)  and \(\mathbb {E}[\Vert {\bar{H}}_i - \nabla ^2f_i\Vert ^2\mid \mathcal {F}_{i-1}]\le \Upsilon _m\) , we know \((\sum _{i=0}^{t-1} {\bar{H}}_i - \nabla ^2f_i)/t\)  is a square integrable martingale.
Thus, [1]} suggests that for any \(\upsilon >0\) ,
\({\frac{1}{t}\sum _{i=0}^{t-1} {\bar{H}}_i - \nabla ^2f_i} = o{\sqrt{\frac{(\log t)^{1+\upsilon }}{t}}}.\) 
